Math You Need > Topographic Maps and Slopes > Slopes

How do I calculate slope/gradient?
"Rise over run" in the geosciences

A hillslope diagram illustrating rise and run. Many of us know that the slope of a line is calculated by "rise over run". However, the application of slope calculation can seem a little more complicated. In the geosciences, you may be asked to calculate the slope of a hill or to determine rate by calculating the slope of a line on a graph. This page is designed to help you learn these skills so that you can use them in your geoscience courses.

Why should I calculate slope or gradient?

In the geosciences slope can play an important role in a number of problems. The slope of a hill can help to determine the amount of erosion likely during a rainstorm. The gradient of the water table can help us to understand whether (and how much) contamination might affect a local well or water source.

How do I calculate slope (or gradient) in the geosciences?

Gradient in the case of hillslope and water table is just like calculating the slope of a line on a graph - "rise" over "run". But how do you do that using a contour (or topographic) map?

map for sample problem in slope explanation
  1. First get comfortable with the features of the topographic map of interest. Make sure you know a few things:
    • What is the contour interval (sometimes abbreviated CI)?
    • What is the scale of the map?
    • What is the feature for which you want to know the slope?
    Below there is a topographic map of Math State Park. You are interested in building a path from the top of the hill on this map to the creek (Equation Creek) and want to know the slope of the hill. You should probably print out the map (with the steps for calculating slope) (Acrobat (PDF) 93kB Oct15 08).
    map for sample problem in slope explanation
    • What is the contour interval of this map?
      The contour interval tells you "rise", specifically the change in elevation between each of the "brown lines" (contours). In this case, the contour interval is in the key in the lower right and is abbreviated CI. The contour interval is 20 ft.
    • What is the scale of the map?
      The scale tells you the "run", or the distance on the ground. On this map, it is also shown in the lower right and is shown only graphically. If you print out the map (with the steps for calculating slope) (Acrobat (PDF) 93kB Oct15 08), you will find that 1 inch = 1 mile.
    • What is the feature for which you want to know the slope?
      In this case, you want to know the slope of the hillside to the WNW from the top (at 869 ft) to the creek.
  2. First, you need to know "rise" for the feature. "Rise" is the difference in elevation from the top to bottom (see the image above). So determine the elevation of the top of the hill (or slope, or water table).
    The top of the hill of interest is 859 ft. The contour line at the creek where your path will end is one below 700 ft. That makes it 680 ft (because the contour interval is 20 ft). The difference in elevation is the top minus the bottom (859 ft - 680 ft) so "rise" = 179 ft
  3. Next you need to know "run" for the feature. "Run" is the horizontal distance from the highest elevation to the lowest. So, get out your ruler and measure that distance. If you know the scale, you can calculate the distance. Most of the time distance on maps is given in km or mi.
    The red line represents the distance along the hillslope where you want to build your path. The red line is twice as long as the scale for one mile (on the printed map, it is about 2 inches). Thus, the distance from the top to bottom of the hill or "run" = 2 mi
  4. Now comes the rise over run part. There are two ways that you may be asked to make calculations relating to slope. Make sure you know what the question is asking you and follow the steps associated with the appropriate process:
    • If you are asked to calculate slope (as in a line or a hillside), a simple division is all that is needed. Just make sure that you keep track of units!
      1. As we've seen throughout this page, slope is "rise over run". The phrase "rise over run" implies that you will need to divide. The equation for slope looks like this:
        rise over run
      2. Take the difference in elevation and divide it by the horizontal difference (always making sure you keep track of units).
        On the map of Math State Park the hill's rise = 179 ft and run is 2 mi. So we set up the problem like this:

        slope calculation example problem
      3. Finish the calculation using your calculator (or doing the calculations by hand).
        Now we just divide the rise by the run and wind up with:

        slope calculation example problem answer

        The units you end up with might be feet/mile or m/km or feet/foot (slope can be expressed in all these ways). It just depends on what you started with.
    • You may also be asked to calculate percent (or %) slope. This calculation takes a couple of steps. And it mostly has to do with paying attention to units. The units on both rise and run have to be the same.
      1. To calculate percent slope, both rise and run must be in the same units(for example, feet or meters). If your horizontal distance is in miles, you need to convert to feet; if the horizontal distance is in kilometers, you'll have to convert to meters. (To convert from miles to feet, multiply by 5280 ft/mi; km to m, multiply by 1000m/km. If you need more help with this or need to convert other units, please see the unit conversions module).
        Right now you have rise in feet and run in miles. Let's convert the miles to feet by multiplying by the appropriate conversion factor: 1 mile = 5280 feet. So, we should multiply "run" by conversion :

        10560 ft
      2. Once you have converted so that both elevation and distance have the same units, we can write an equation for slope: rise over run (implying rise divided by run).
        We know that rise is 179 ft and run is 10560 ft:

        rise run % slope sample

        But, hold on, there's one more step to getting to % slope.
      3. To get to % we need to multiply the calculated slope (which is unitless because the units cancel on the top and bottom) by 100 so that our equation looks like this:
        % slope equation
        Start with rise over run and multiply by 100:
        set up % slope sample
      4. Now plug in your numbers and calculate % slope!
        cancel % slope sample
        Note that % slope does not have any units because ft cancel in the calculation. Make sure that you indicate that it's % however!

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